Study Guide

Exponential and Logarithmic Functions - Logarithmic Functions

Logarithmic Functions

Save the trees! This section is all about logarithms: converting exponentials to logarithms, evaluating logarithms, and the properties of logarithms that will help you in solving equations later. This is a super important section, so settle in. Get a cup of joe, sit back, relax, and enjoy the show.

  • The Basics

    The Logarithm: a secret dance that the trees do when we are not looking.

    Actually, the logarithm is far more complicated than that. You have learned about logarithms before, but just to catch you up, here are the basics. Logarithms are inverses of a number to a power. The general equation for a logarithmic function is:

    y = logbx

    This actually means x is the answer (or the logarithm) to b raised to the y power or:

    x = by

    b is the base, so when we are talking about logarithms, b is the base of the exponent.

    Sample Problem

    Show 2 × 2 × 2 × 2 × 2 × 2 in logarithmic and exponential form.

    2 × 2 × 2 × 2 × 2 × 2 = 64
    log264 = 6
    26 = 64

    Sample Problem

    Okay, let's try to evaluate a logarithm:

    Evaluate log101000. So we ask ourselves 10 to what power is 1000?

    103 = 1000
    log101000 = 3

    In words, we would say the logarithm is 3. What we have really been talking about here is an inverse. The inverse of the logarithm of base 10 is 10 to the logarithm. We will talk about inverses of the logarithm graphs in the next section but it is helpful to understand what log really means.

    Sample Problem

    Evaluate the logarithm:

    log5. So we ask ourselves 5 to what power is ? Remember the square root is a number to the  power.

    Sample Problem

    Evaluate this:

    3log37. 3 is the base, and 3 is the base in the logarithm, so because they are inverses, the answer is 7! How easy was that?

    Sample Problem

    Evaluate:

    log10102.

    Again, we are applying the inverses here since the log with base 10 is the inverse of an exponent with base 10. The answer is 2. Badabing-badabang.

    Sample Problem

    Evaluate log 235.

    Wait, where is the base? If there is not base, that means it is a base of 10. Logarithm with base 10 is called a common logarithm. It would be hard to figure out 10 to what power is 235. It will end up being a decimal. We will have to evaluate this logarithm on your calculator!

    We are not sure what calculator you have but if you have a scientific calculator, usually the log function thinks you are in base 10. Find the log button on your calculator and type log 235. The answer rounded to the hundredth is: 2.37.

    If you want to check this, take 2.3710 which is 234.423. This is not 35 because we rounded. It would be more accurate if we checked this way: 102.371067862 which is very close to 235.

    Why even use logarithmic functions? We have used these for centuries before computers came along and it was to help us multiply large numbers or take the powers of numbers. For instance, instead a calculator, there were tables of logarithms. We don't need old tables of logarithms or slide rules (old-fashioned calculators for old people), we can use calculators. Goodbye 18th century.

    There are a few properties you have learned before in Algebra 2, but as a review, here they are:

    Properties of Logarithms

    Product Rule: logxab = logxa + logxb
    Quotient Rule:
    Power Rule: logxab = blogxa

    Sample Problem

    Rewrite log 3ab3 as a sum of logarithms with no exponents:

    log3ab3 = log(3) + log(a) + log(b3)
    = log(3) + log(a) + 3log(b)

  • Graphs of Logarithmic Functions

    Imagine your world flipped upside down and backwards. That's what happened to the exponential function, and in this section we are exploring the inverse of an exponential function...drum roll please...The Graph of the Logarithmic Function. We're talking about the graphs of logarithmic functions, and how they have a vertical asymptote (compared to a horizontal one in exponential functions). We won't forget the good stuff like domain and range. We'll also give you a few tips of graphing these on technology.

    Since we have already decided that logarithms were the inverses of the power function, we can see that when we graph these, they are symmetrical about the diagonal line y = x which makes a 45 degree angle with the origin. Huh?

    Sample Problem

    Graph y = 10x, y = log10x, and y = x for values greater than 0.

    Okay, here's a play by play of the above graph:

    • The red line is y = 10x.
       
    • The blue line is y = log10x.
       
    • The green line is y = x.
       
    • The red line and blue line don't like each other.

    Every function's inverse is a reflection over the diagonal line y = x. To think of this in the real world, picture yourself leaning on a counter at a 45 degree angle and looking into the mirror. If you're lucky, you might be staring directly at your inverse. Or you may realize you need to clean the mirror.

    Logarithmic graphs are just plain weird. They are different, for one thing, because they have a vertical asymptote rather than a horizontal asymptote.

    Sample Problem

    Graph y = log2x for x > 0.

    If you have a graphing calculator, you could plot this in your y-editor but you might have to convert the base 2 logarithm into base 10. Here's how you do that:

    Instead of using:

    y = log2x

    Use this:

    In fact, any time you get the fancy to convert a logarithm into something you can use on the calculator, simply divide the log x function (which is naturally base 10) by the log of the base that you are in.

    Sample Problem

    Graph y = log4x and y = log40x and explain the differences:

    We will graph  and  so each function will be in the same base (common logarithm which is base 10).

    Here are some qualities of this graph above

    • The red function is y = log40x.
        
    • The blue function is y = log4x.
       
    • The graph looks odd because we are showing you negative x values too, which happen to be imaginary values (we usually don't graph these) but we thought it would be fun to look at.
       
    • The vertical asymptote is y = 0 for each function.
        
    • Domain for each function: (0,∞), we didn't include the negative x values because the y values are not real numbers.
       
    • Range for each function: All real numbers.
       
    • Continuous? Yes, only if you are consider values of x > 0.
       
    • Is the function increasing or decreasing? increasing if we just look at positive x values.
       
    • Vertical asymptote: y = 0.
       
    • Loves listening to Stephen Hawking and is fascinated with black holes.
  • The Natural Log

    You are about to learn the single most important concept in solving exponential and logarithmic equations. The natural log is a very handy tool to keep in your mathematical tool belt in this chapter. The natural log is not only the inverse of the ex function, but it is used directly in later sections to solve both exponential and logarithmic equations.

    You will look at the graphs of the natural log function, practice using the properties, and also evaluate natural log functions on your calculator. Don't think too hard about making jokes about the natural log, we already know about them but we'll save you from the pain.

    The natural log, ln, is a special case where you have f(x) = loge x. The e is the base! If we wrote it out the long way, it would look like this: f(x) = log2.718281828... x. In 1668, some bright guy named Nicholas Mercator decided to call it the natural log function. In the 1600's, we didn't have bloggers, we had loggers, also known as old fashioned nerds.

    These are are equivalent statements:

    f(x) = loge x
    f(x) = ln(x)

    Notice how the base of the logarithm is e. We call this the natural log so we don't have to write loge. We can simply write y = ln(x) and everybody understands what that stands for. Our little friend, e, wanted to have his own logarithmic function. What a guy!

    Sample Problem

    Graph the natural log function and it's inverse.

    The blue curve is the y = ex function, and the natural log function y = ln(x) is the red curve. Since they are inverses, they are basically a reflection over that imaginary diagonal line y = x

    Sample Problem

    Using the properties of logarithms (from our earlier section), rewrite this natural log function as a sum or a difference of logarithms:

    Rewrite  in sums and differences of natural logarithms.

    Okay let's do this in steps, first since it's a natural log, it follows the same properties of logarithms (which we learned in an earlier section).

    Speaking of inverses, did you know that we could apply the inverse of the natural log to the exponential function? Watch this magic:

    elnx = x
    ln(ex) = x

    Sample Problem

    Evaluate eln9.

    Since we now know that the inverse of an e function is ln, the solution is:

    eln9 = 9.

    Sample Problem

    Evaluate ln(e1.9).

    Easy Shmeazy. The answer is 1.9 thanks to our trusty friend, the inverse.

    In the last section, Graphs of Logarithmic Functions, we mentioned that you could convert a logarithm of a certain base to a logarithm of base 10 with a simple trick.

    Remember this...

    y = log2 x

    ...is the same as this:

    Similarly, we can do this with natural logs which will help us in the next section solve equations. If we have y = log3 x, we can convert it to a natural log function, which some people seem to like better than working with logs.

    Natural logs are less cumbersome and there are nifty little buttons on your calculator that are for ln and you don't have to worry about changing bases. Break out the granola, we're going natural!

    Sample Problem

    Convert log6 100 to natural logs and then evaluate (round to nearest hundredth).

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